On Happyness and Generators of Groups

It usually happens to me. There are days when all I can think about is completing the work at hand or about my course of action for coming days. So I just keep on reading stuff, finishing assignments, preparing for exams, working on some of my ideas, dreaming and day-dreaming about rosy career-prospects ;-) ... and then somewhere in the middle of it, I realize as if I am trying to run as fast as I can without enjoying what the 'present' has to offer to me. It doesn't look very good and then I try to take a sort of small break and as a result some funny ideas come to my mind. So, this is a recollection of one of the recent such moments I had.

I was trying to learn some functional programming few minutes before this break that I mentioned. I pondered over the reason of why I wanted to learn functional programming. I answered, "because that would make me a better programmer". Okay, but why do I want to be a better programmer ? I answered, "because I love programming and it's really cool to know that you have learned something important, you have enhanced your skill, you have a wider perspective". Fine, but then I questioned, why does it matter really that I have gained a broader vision about programming a solution to a problem. To cut the chase, it finally dawned on me that the reason was "Happyness" (yes, that spelling is borrowed from the movie). Moreover, it dawned on me that in fact, Happyness was the answer if you started to ask questions and reduce your pursuits to some fundamental reason. I am very sure on this that everyone of us can apply this method of reasoning to themselves and the answer will inevitably turn out to be Happyness. It may be studying maths, working hard on your tennis games, practicing new moves in chess, staying late at office to finish some task ;-) ... whatever.

Well, that's half the story. A remarkable characteristic of human mind is that it's working is strongly based on patterns and it learns a lot through drawing analogies. Surprisingly, a strange analogy struck me in regards to the above mentioned stream of thought. Let me now bring you to the topic of groups and generators which will help you understand my crazy analogy.

Algebra is a beautiful subject indeed. It represents an example of what the mainstream mathematics of today looks like, which is, abstract. Yes, modern mathematics is hugely abstract in nature. It may surprise some people as mostly it is thought to be involving calculations. In fact, this was the scene a few centuries back. However, over the course of time, the emphasis has moved from computing to understanding, which brings in abstractness. Don't misinterpret me in that there was no understanding involved in mathematics a few centuries back. It's just that mathematics then was mostly used as a tool to calculate answers to problems, mostly physics-related. What happened that mathematicians studied some related problems and then in the process of study, refined their results more and more to some basic essentials, which became the abstract theories of mathematics we have today. In Algebra, we study mainly algebraic systems (such as groups, rings, fields), which simply put are, a set of objects together with some operations for combining them.

Which brings us to Groups. Now, I am not going to bomb you with heavy mathematical definitions. I will try to explain to you in as much a layman-style as possible. Consider the set Z11={1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Now, we will add an operation to this set ... it's multiplication (and you thought it was going to be some Lebesgue integral over the norms of the p-adic variation of this group !). Yes, plain old multiplication but with a little twist, it's called multiplication modulo 11 (in general modulo n). This is nothing complicated, all you have to do is that whenever you multiply any two numbers in the set Z11, you take the remainder of product divided by 11 as your result. For example, 2 x 6 = 1, because you take the remainder of 12 divided by 11. Similarly, 3 x 5 = 4, 4 x 7 = 3 and so on. Guess what? We have a group at our hands and it is the pair (Z11, multiplication modulo 11). Technically, there are some other properties this pair needs to satisfy to be called a group but, I won't be going into those and you can believe me here in that this pair is indeed a group.

So, with the group at our hands, it's time to look at the element 2 in the set Z11. 2 here has a special property as does the group Z11. I am going to denote power operation by ^. Observe now. 2^0=1, 2^1=2, 2^2=4, 2^3=8, 2^4=5 (not 16, remember we divide the result of multiplication by 11), 2^5=10, 2^6=9, 2^7=7, 2^8=3, 2^9=6, and 2^10=1 again. What we have here is that a single element (2 here) generates the whole set when the operation is applied to it repeatedly. The power thing is actually 2 being multiplied with itself again and again. Thus, 2 is special and it is called the generator of the group (Z11 itself is special since it is an example of groups that have generators, many groups don't ... in fact put any prime in the place of 11, say p, and you will see that 2 is a generator of Zp).

Okay, let me get to my crazy idea that I was going to talk to you after all. Probably you might have already got it or a feeling of what it is. To kill all the hyped suspense, the idea is:

Happyness is the generator of the "group" formed over the pursuits or actions in our life.

Crappy, isn't it ? Yeah, sorry to bother you all along but let me give some reasons as to why I think so. First, you can see the strong analogy between the questioning process I mentioned and the generation of group by a generator. What I mean to say is that, each of your actions fundamentally derives from the a desire to be happy. Hence, happyness when applied to itself through an operation (that I am not very clear about) generates all your other desires, actions and pursuits. Secondly, sometimes you find that at some stage of your questioning you may have more than one reason/answer for your question. This can also be explained in the analogy if you observe that 4^2=5, 4^3=9, 4^4=3. So 4 also generates some elements but not the whole group (try it, you will know why, it's not tough). It is 2 that generates the whole group. So is it with Happyness, it is the "fundamental" generator, or in other words, the generator of the (whole) group.

Now, you may pose a valid question ... if we consider one person's desire/actions to be a group and do the same for another person, is it the case that the happyness element that they both have in their groups are the same ? I mean to say that whether when you feel happy and I feel happy (when the happyness element is expressed), is it the same feeling ? Well, I should say that analogies when carried too far or stretched too much, usually break. So, I am doubtful if this one holds. On a different note, this question is a very important one in Algebra, as it is basically the question of two groups being isomorphic (and I don't have the time to explain that here, maybe some other post :-)

"This person has gone nuts!" ... must be the thought coming to your mind, so let me say goodbye and thanks for reading.

The Question of Which Tire & Focal Point Strategy

There's a mail that has been circulated a lot in student groups as well as among people in software companies (as I came to witness during my software job).

This is about two friends who had a very good preparation for Chemistry (whatever) final for the semester. However, they had a party on weekend to attend and they decided to go for it. The party went really well and they missed to reach for the final on time. So, they concocted a story about their car tire being punctured and presented it to the professor as the reason why they were unable to reach on time. They asked for a makeup final and the professor agreed. When the final came, they were made to sit in different rooms and had two questions in the paper. First one was quite easy, 10 marks scored and they thought it was going to be cool when they turned the paper and found the second question, "Which tire ?".

Now this incident is claimed to be real and that it took place in IIT-Bombay. However, I doubt that this is true since I have been reading Game Theory (you must have heard this name if you have watched the movie, A Beautiful Mind) from a very nice book (Games of Strategy - Avinash Dixit, Susan Skeath) and there I found this to be a Duke University incident. Anyways, that's not the interesting stuff. What makes this thing interesting is that it presents an example of what are called focal point strategies.

In fact, there is one more strategic lesson from this story. That is, to recognize that professor is an intelligent (rational, in game theory jargon) player in this game. The students failed to take into account that professor may suspect their story to be fake, and that he may come up with such a question (whether in the final or maybe before it).

Now, coming back to the focal point strategies. The probability that both the friends picked up the same tire is easy to calculate and it is 1/4 or 25% (4 times 1/4 times 1/4). So they need to come up with some reasoning besides pure luck. Say, you are one of the students and you have some reasons to believe that the front right tire is most likely for puncture (pointed objects are more likely to be on side of the road, the trip had right turns mostly or anything you can come up with). What makes your choice good is not the reasons which you have thought of but, whether your friend has also made the same choice. So you need to consider whether your friend would think the same way (and pick the same choice as yours) and, that he thinks that you will also think the same way that he thinks that you think ... and so on. A chain of reasoning, and it needs to converge for the outcome to be in your favour (and your friend's). Hence, the name focal point. So to quote from the book, "what is needed is a convergence of expectations about what should be chosen in such circumstances. Such a commonly expected strategy on which players can successfully coordinate is called a focal point."

There is obviously more than this to focal points but, for starters it is quite a good example. I will be putting some more as I keep learning from the book.